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Theorem relexpind 25141
Description: Principle of transitive induction, finite version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
relexpind.1  |-  ( et 
->  Rel  R )
relexpind.2  |-  ( et 
->  R  e.  _V )
relexpind.3  |-  ( et 
->  S  e.  _V )
relexpind.4  |-  ( et 
->  X  e.  _V )
relexpind.5  |-  ( i  =  S  ->  ( ph 
<->  ch ) )
relexpind.6  |-  ( i  =  x  ->  ( ph 
<->  ps ) )
relexpind.7  |-  ( i  =  j  ->  ( ph 
<->  th ) )
relexpind.8  |-  ( x  =  X  ->  ( ps 
<->  ta ) )
relexpind.9  |-  ( et 
->  ch )
relexpind.10  |-  ( et 
->  ( j R x  ->  ( th  ->  ps ) ) )
Assertion
Ref Expression
relexpind  |-  ( et 
->  ( n  e.  NN0  ->  ( S ( R ^ r n ) X  ->  ta )
) )
Distinct variable groups:    x, n    et, x, i, j    S, i, x, j    R, i, x, j    ps, i,
j    th, i    ch, i    ph, j, x    x, X    ta, x
Allowed substitution hints:    ph( i, n)    ps( x, n)    ch( x, j, n)    th( x, j, n)    ta( i, j, n)    et( n)    R( n)    S( n)    X( i, j, n)

Proof of Theorem relexpind
StepHypRef Expression
1 relexpind.4 . 2  |-  ( et 
->  X  e.  _V )
2 relexpind.1 . . . . 5  |-  ( et 
->  Rel  R )
3 relexpind.2 . . . . 5  |-  ( et 
->  R  e.  _V )
4 relexpind.3 . . . . 5  |-  ( et 
->  S  e.  _V )
5 relexpind.5 . . . . 5  |-  ( i  =  S  ->  ( ph 
<->  ch ) )
6 relexpind.6 . . . . 5  |-  ( i  =  x  ->  ( ph 
<->  ps ) )
7 relexpind.7 . . . . 5  |-  ( i  =  j  ->  ( ph 
<->  th ) )
8 relexpind.9 . . . . 5  |-  ( et 
->  ch )
9 relexpind.10 . . . . 5  |-  ( et 
->  ( j R x  ->  ( th  ->  ps ) ) )
102, 3, 4, 5, 6, 7, 8, 9relexpindlem 25140 . . . 4  |-  ( et 
->  ( n  e.  NN0  ->  ( S ( R ^ r n ) x  ->  ps )
) )
1110ax-gen 1556 . . 3  |-  A. x
( et  ->  (
n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ps ) ) )
1211a1i 11 . 2  |-  ( et 
->  A. x ( et 
->  ( n  e.  NN0  ->  ( S ( R ^ r n ) x  ->  ps )
) ) )
13 id 21 . 2  |-  ( et 
->  et )
14 relexpind.8 . . . 4  |-  ( x  =  X  ->  ( ps 
<->  ta ) )
15 breq2 4217 . . . . . . . 8  |-  ( x  =  X  ->  ( S ( R ^
r n ) x  <-> 
S ( R ^
r n ) X ) )
1615imbi1d 310 . . . . . . 7  |-  ( x  =  X  ->  (
( S ( R ^ r n ) x  ->  ta )  <->  ( S ( R ^
r n ) X  ->  ta ) ) )
1716imbi2d 309 . . . . . 6  |-  ( x  =  X  ->  (
( n  e.  NN0  ->  ( S ( R ^ r n ) x  ->  ta )
)  <->  ( n  e. 
NN0  ->  ( S ( R ^ r n ) X  ->  ta ) ) ) )
1817imbi2d 309 . . . . 5  |-  ( x  =  X  ->  (
( et  ->  (
n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ta ) ) )  <->  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) X  ->  ta ) ) ) ) )
19 imbi2 316 . . . . . . . 8  |-  ( ( ps  <->  ta )  ->  (
( S ( R ^ r n ) x  ->  ps )  <->  ( S ( R ^
r n ) x  ->  ta ) ) )
2019imbi2d 309 . . . . . . 7  |-  ( ( ps  <->  ta )  ->  (
( n  e.  NN0  ->  ( S ( R ^ r n ) x  ->  ps )
)  <->  ( n  e. 
NN0  ->  ( S ( R ^ r n ) x  ->  ta ) ) ) )
2120imbi2d 309 . . . . . 6  |-  ( ( ps  <->  ta )  ->  (
( et  ->  (
n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ps ) ) )  <->  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ta ) ) ) ) )
2221bibi1d 312 . . . . 5  |-  ( ( ps  <->  ta )  ->  (
( ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ps ) ) )  <->  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) X  ->  ta ) ) ) )  <->  ( ( et  ->  ( n  e. 
NN0  ->  ( S ( R ^ r n ) x  ->  ta ) ) )  <->  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^ r n ) X  ->  ta )
) ) ) ) )
2318, 22syl5ibr 214 . . . 4  |-  ( ( ps  <->  ta )  ->  (
x  =  X  -> 
( ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ps ) ) )  <->  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) X  ->  ta ) ) ) ) ) )
2414, 23mpcom 35 . . 3  |-  ( x  =  X  ->  (
( et  ->  (
n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ps ) ) )  <->  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) X  ->  ta ) ) ) ) )
2524spcgv 3037 . 2  |-  ( X  e.  _V  ->  ( A. x ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
r n ) x  ->  ps ) ) )  ->  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^ r n ) X  ->  ta )
) ) ) )
261, 12, 13, 25syl3c 60 1  |-  ( et 
->  ( n  e.  NN0  ->  ( S ( R ^ r n ) X  ->  ta )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2957   class class class wbr 4213   Rel wrel 4884  (class class class)co 6082   NN0cn0 10222   ^ rcrelexp 25128
This theorem is referenced by:  rtrclind  25150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-seq 11325  df-relexp 25129
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